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Dini's Theorem

Dini's theorem is a result in real analysis relating pointwise convergence of sequences of functions to uniform convergence on a closed interval.

Let F={f_n} be a sequence of real-valued continuous functions on a closed interval I=[a,b]. Suppose that, for every x in I, the sequence {f_n(x)} is an increasing sequence and converges to f(x), where f is a continuous function on I. Then Dini's theorem states that F converges to f uniformly on I.

See also

Continuous, Continuous Function, Convergent Sequence, Increasing Sequence, Interval, Pointwise Convergence, Uniform Convergence

This entry contributed by Christopher Stover

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References

Royden, H. L. and Fitzpatrick, P. M. Real Analysis, 4th ed. Boston, MA: Pearson-Prentice Hall, 2010.

Referenced on Wolfram|Alpha

Dini's Theorem

Cite this as:

Weisstein, Eric W., with contributions by Christopher Stover. "Dini's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DinisTheorem.html

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